Let $X$ and $Y$ be two locally compact Hausdorff spaces
i. Show that each $f \in C_c(X \times Y)$ is a limit of sums of the form $$\sum\limits_{i=1}^n \varphi_i(x) \psi_i(y)$$ where $\varphi_i \in C_c(X)$ and $\psi_i =C_c(Y)$
ii. Show that $\mathcal{B}(X \times Y) \subseteq \mathcal{B}(X) \times \mathcal{B}(Y)$
iii Show that $\mathcal{B} (X \times Y) = \mathcal{B}(X) \times \mathcal{B}(Y)$ if and only if X or Y is the union of a countable collection of compact subsets.
On i. we can use Stone-Weierstrass Theorem. Here is some ideas for ii. but as you can see it gets complicated and there might be another way to do this. Also any help on iii. is greatly appreciated.
ii.
Let $f(x,y)= \varphi(x) \cdot \psi(y)$ and $$A = \{(x,y) \ | \ f(x,y) >c \} = \{ (x, y) \ | \ \varphi(x) \cdot \psi(y) > c \}$$ Also consider $$A_q = \{(x,y) \ | \ \varphi(x) > R_x, \psi(y) > R_y \}=\{x \ | \ \varphi(x) > R_x \} \times \{y \ | \ \psi(y) > R_y \}$$ where $R_x$ and $R_y$ are $x$ and $y$-section of $f$ We have: $$A= \bigcup \{x \ | \ \varphi(x) > p \} \times \{y \| \ \psi(y) > q \}$$ where $$(p,q) \in Q \times Q \cap \{(x,y) \ | xy >c\}$$
Let $f(x,y) = \varphi_1(x) \cdot \psi_1(y) + \varphi_2(x) \cdot \psi_2(y)$ $$\{(x,y) \ | \ f(x,y) > c \} = \{(x,y) \ | \ \varphi_1(x) \cdot \psi_1(y) + \varphi_2(x) \cdot \psi_2(y) >c \}$$
For (i), consider a subalgebra of $C_c(X\times Y)$: $$\mathcal{A}=\{h(x,y)|h(x,y)=\sum_{i=1}^nf_i(x)g_i(x),n\in\mathbb{N^+},f_i\in C_c(X),g_i\in C_c(Y)\}$$ then by stone-weierstrass theorem, $\mathcal{A}$ is dense in $C_c(X\times Y)$, which proves (i).
For (ii), By math induction.